There are two polynomials \(A, B \in  {\mathbb {F}}_q[x]\)  such that \(Af+Bg=Res(f,g,x)\) , and so \(Res(f,g,x)=0\)  if and only if \(f(x)\)  and \(g(x)\)  have a common divisor of positive degree.
For two polynomials \(F(x,y),\, G(x,y)\in \mathbb {F}_q[x,y]\)  of positive degree in \(y\) , the resultant \(Res(F,G,y)\)  with respect to \(y\)  is a polynomial in \(x\) , and is in the elimination ideal
\(\langle F, G\rangle \cap  {\mathbb {F}}_q[x]\) . So, \(Res(F,G,y)\)  vanishes at any common solution of \(F=G=0\) . For more information on resultants, the reader can refer to [1]}.
